Question

use the special triangles on the unit circle to determine \theta in degrees when \sin \theta = \frac{\sqrt{3}}{2}.

Explanation:

Step1: Recall the definition of sine in the unit circle

In the unit circle, for a point \((x,y)\) on the terminal side of an angle \(\theta\), \(\sin\theta=y\).

Step2: Locate the \(y -\)coordinate

We are given \(\sin\theta=\frac{\sqrt{3}}{2}\). Looking at the points on the unit - circle diagram \((\frac{1}{2},\frac{\sqrt{3}}{2})\) and \((\frac{\sqrt{3}}{2},\frac{1}{2})\), the \(y -\)coordinate \(\frac{\sqrt{3}}{2}\) corresponds to the angle whose terminal side passes through the point \((\frac{1}{2},\frac{\sqrt{3}}{2})\).

Step3: Identify the angle

The angle \(\theta\) whose terminal side passes through the point \((\frac{1}{2},\frac{\sqrt{3}}{2})\) is \(60^{\circ}\) (since for the \(30 - 60-90\) triangle in the unit circle, when the side opposite the angle is \(\frac{\sqrt{3}}{2}\) (in the first - quadrant unit - circle context), the angle is \(60^{\circ}\)).

Answer:

\(60^{\circ}\)